Singular McKean-Vlasov (Reflecting) SDEs with Distribution Dependent Noise

TL;DR

This paper establishes well-posedness and regularity for singular McKean-Vlasov SDEs with distribution-dependent noise using Zvonkin's transformation and a fixed point approach, extending previous results to more complex noise structures.

Contribution

It introduces a novel method combining Zvonkin's transformation with a two-step fixed point argument to handle singular McKean-Vlasov SDEs with complex distribution-dependent noise.

Findings

Proved well-posedness for a class of singular McKean-Vlasov SDEs.

Derived regularity estimates for solutions.

Extended results to singular reflecting SDEs with distribution-dependent noise.

Abstract

By using Zvonkin's transformation and a two-step fixed point argument in distributions, the well-posedness and regularity estimates are derived for singular McKean-Vlasov SDEs with distribution dependent noise, where the drift contains a term growing linearly in space and distribution and a locally integrable term independent of distribution, while the noise coefficient is weakly differentiable in space and Lipschitz continuous in distribution with respect to the sum of Wasserstein and weighted variation distances. The main results extend existing ones derived for noise coefficients either independent of distribution, or having nice linear functional derivatives in distribution. Singular reflecting SDEs with distribution dependent noise are also studied.